A remark on split least-squares mixed element procedures for pseudo-parabolic equations☆
Introduction
We consider the following initial-boundary value problemwhere J = (0, T] is the time interval, Ω is a bounded polygonal domain in Rd(d = 2, 3), with a Lipschitz continuous boundary Γ. f = f(x, t) is a given function. We shall make the following assumptions on the coefficients c, a and b: there exist positive constants c∗, c∗, a∗, a∗, b∗, b∗ and k1 such that
The above equations are usually called pseudo-parabolic or Sobolev type equations, which appear in engineering fields such as, for instance, flows of fluids through fissured rock, heat conduction involving a thermodynamic temperature and a conductive temperature, and quasi-stationary processes in semiconductors (see, e.g. [1], [2]). For a discussion of existence and uniqueness results, see Refs. [3], [4]. Various finite element schemes have been constructed to treat such problems, see [5], [6]. Mixed methods to approximate such problems have been formulated in [7].
We have introduced an elegant theory of the least-squares methods for such equations in [8]. The least-squares mixed element procedure has two typical advantages as follows: it is not subject to the Ladyzhenskaya [9] – Babuska [10] – Brezzi [11] consistency condition, so the choice of approximation spaces becomes flexible, and it results in a symmetric positive definite system. The method has been introduced for elliptic problems [12], and time dependent problems [13], [14]. However, the conventional least-squares finite element procedure usually needs to solve a coupled system of equations, which leads to difficulties in some cases.
Recently, in [15], [16] a kind of split least-squares Galerkin procedure was constructed for stationary diffusion reaction problems and parabolic problems. In this paper, we apply this idea and give two novel split least-squares mixed element procedures to solve the pseudo-parabolic equations. By selecting the least-squares functional properly, the resulting least-squares mixed element procedures can be split into two symmetric positive definite sub-schemes, one of which is for the primary unknown variable u and the other is for the introduced unknown flux variable σ. In the first procedure, the sub-procedure for u is the same as the standard Galerkin finite element procedure with first-order approximation in time increment. In the second procedure, the sub-procedure for u is the same as the Crank–Nicolson standard Galerkin finite element procedure with second-order approximation in time increment. The convergence analysis shows that both methods lead to the optimal order H1(Ω) and L 2(Ω) norm error estimates for the primal unknown u and optimal H(div; Ω) norm error estimate for σ. Finally, we give a numerical example which is in good agreement with the theoretical analysis.
The paper is organized as follows. In Section 2 we formulate the split least-squares mixed element procedures. The convergence theory on the novel algorithms is established in Section 3. In Section 4 we give the numerical experiment.
In this paper we use Wk,p(k ⩾ 0, 1 ⩽ p ⩽ ∞) to denote Sobolev spaces [17] defined on Ω with usual norms and Hk(Ω),L2(Ω) with norms . For simplicity we also use Ls(Hk) to denote Ls(0,T; Hk(Ω)). The inner product (·, ·) is both used for scalar-valued functions and vector-valued functions. Throughout this paper, the symbols K and δ are used to denote a generic constant and a generic small positive constant respectively, which may appear differently at different occurrences.
Section snippets
Split least-squares mixed element procedures
Introduce two function spaceswith norm
Introducing the flux variable σ = −(a∇ut + b∇u), we can rewrite problem (1.1) as a first-order system
Given a time step Δt = T/N, where N is a positive integer, we shall approximate the solution at times tn = nΔt, n = 0,1 ,… , N. Let un(x) = u(x,tn) and .
By using the difference technique with
Convergence analysis
In this section, we analyze the convergence of the two procedures. From the approximate property of finite element spaces we know that for a given there exists a vector-valued function such that
Now we consider the error estimate for Scheme I. Theorem 3.1 Let (σ, u) be the solution of system(2.1) and be the solution of Scheme I. When the solution (σ, u) is sufficiently smooth and Δ t, hu and hσ are sufficiently small, there hold the
Numerical example
In this section, we give a numerical example to show the efficiency of the presented schemes. We consider the following problem in a two-dimensional domainwhere Ω = (0, π] × (0, π], a, b are constants, . b > 0 is the diffusion constant. The term ∇·(a∇ ut) is interpreted as due to the viscous relaxation effects. The analytical solution of (4.1) is . Then
Conclusion
We have presented two novel split least-squares finite element procedures for pseudo-parabolic equations. By selecting the least-squares functional properly, the resulting procedures can be split into two symmetric positive definite sub-schemes, one of which is for the primary unknown variable u and the other is for the introduced unknown flux variable σ. We have proved the methods yield optimal estimates in the corresponding norms. The numerical results given above are in good agreement with
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